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of algebra advocated by his British contemporaries, Hamilton
sought to found algebra on something 'more reaF, and for this
strictly meaningless enterprise he drew on his knowledge of
Kant's mistaken notions - exploded by the creation of non-
Euclidean geometry - of space as "a pure form of sensuous
intuition'. Indeed Hamilton, who seems to have been unac-
quainted with non-Euclidean geometry, followed Kant in
believing that 'Time and space are two sources of knowledge
from which various a priori synthetical cognitions can be
derived. Of this, pure mathematics gives a splendid example in
the case of our cognition of space and its various relations. As
they are both pure forms of sensuous intuition, they render
synthetic propositions a priori possible/ Of course any not
utterly illiterate mathematician to-day knows that Kant was
mistaken in this conception of mathematics, but in the 1840*s,
when Hamilton was on his way to quaternions, the Kantian
philosophy of mathematics still made sense to those - and they
were nearly all - who had never heard of Lobatchewsky. By
what looks like a bad mathematical pun, Hamilton applied the
Kantian doctrine to algebra and drew the remarkable conclu-
sion that, since geometry is the science of space, and since time-
and space are "pure sensuous forms of intuition', therefore the
rest of mathematics must belong to time, and he wasted much
of his own time in elaborating the bisarre doctrine that algebra
is the science of pure time.
This queer crotchet has attracted many philosophers, and
quite recently it has been exhumed and solemnly dissected by
owlish metaphysicians seeking the philosopher's stone in the
gall bladder of mathematics. Just because 'algebra as the
science of pure time' is of no earthly mathematical significance,
it will continue to be discussed with animation till tune itself
ends. The opinion of a great mathematician on the 'pure time'
aspect of algebra may be of interest. "I cannot myself recognize
the connexion of algebra with the notion of tune,' Cayley con-
fessed; 'granting that the notion of continuous progression
prestnts itself and is of importance, I do not see that it is in
any wise the fundamental notion of the science.9
Hamilton's difficulties in trying to construct an algebra of